The Soft Sell on Hard Sails
The Soft Sell on Hard Sails
With Sail America winning the America's cup challenge with a rigid sail, you can bet that wings for propelling sailboats are about to become more popular, if not a fad. I've always known that solid sails were more efficient than a bent rag, but the Old Salts I have tried to talk into trying one have been skeptical. Now, with success in a big-time race, it may be time to dance a new hornpipe.
It turns out that it doesn't take a lot of theory to show why a solid sail goes better than the traditional cloth ones. In fact, a few multiplications are all that is necessary to demonstrate the point. While I hope and trust that the beauty of the time-honored flexible sail will never leave the seas, at the same time, I also hope that our love of performance will bring increasing numbers of rigid sails to our boats.
[Figure 0, 10-foot dingy with rigid sail]
The purpose of a sail is to generate a horizontal force that propels the boat. For a given weight and size sail, you want to generate as much force as possible. This is why we spend a lot of time making our sails as effective as we can.
Unfortunately, except for very small boats (models, actually) and at very low windspeeds (under 1 meter/second for a ten-foot dingy) the usual sail performs quite badly from a driving force point of view, and we can do a lot better.
AERONAUTICAL TERMINOLOGY
Since the subject here is aerodynamics, we use some aerodynamic rather than nautical terms. For example, if a pirate boarded your craft and took a horizontal swipe with his cutlass, slicing mast and sail in one go while you were under way, and you looked straight down on the lower part in the instant before it all collapsed, you would see a cross section of the mast and sail. That cross section would show a round (or oval, or what have you) mast attached to a thin, curved line. That thin, curved line— your sail — is an airfoil. The mast is its leading edge, so called because the wind always goes around the mast first before playing over the sail itself.
[Figure 1]
If that same pirate were to slice an airplane wing, you would see a shape that looks more like a fish when seen from the top (ignoring its fins).
[Figure 2]
In these shapes lies an interesting tale, which begins around the turn of the century.
WHEN THE WRIGHTS WERE WRONG
After winning bicycle races and while on their way to inventing the airplane, the Wright brothers invented the aileron and the wind tunnel. The aileron is what allows airplanes to make safe, efficient banked turns. The wind tunnel allowed them to test their ideas on small models without having to risk life and fortune on larger, man-carrying craft. This led them to make one of their few mistakes. They wanted to know what airfoil cross-section was best. They tried many but found that a thin, curved plate — very much like a sail — gave the most lift for a given area. And so they built the wings of their planes that way. Lift in airplanes with horizontal wings is pointed upward, but in boats with their vertical "wings" it is aimed horizontally. Lift is what makes sailboats go (excepting when they are running directly downwind).
The Wrights' planes would have flown better if they hadn't used the thin curves their wind tunnel predicted. It wasn’t long before designers understood this, and by the 1920's aircraft wing airfoils used much thicker shapes. The Wrights weren’t dumb: at the size and airspeeds they could test in their tunnel, the thin, curved airfoil is best! What they didn’t know was that about twenty years earlier the English scientist Osborne Reynolds had discovered that (in standard sea-level air) two geometrically similar but different size wings perform the same only if the product of the length of each wing and the velocity of the air going by them is the same.
So a model wing that is 1/10th the size of the actual one must be tested at 10 times the actual air velocity to get correct results. Reynolds's formula also involved viscosity and density so that one can compare performance in water to that in air, and so on, but since we are only talking about ordinary air at average temperatures and pressures, we can ignore the other factors here. Reynolds invented a number, now called the Reynolds number, Re. If this number is the same for two objects, then the two objects will perform similarly in terms of their fluid dynamics. If you measure air velocity in meters per second, and the length of an object in meters, than Re = 70,000 * velocity * length.
There is a much-reproduced classic illustration of the effectiveness of different airfoil shapes at various Reynolds number (e.g. R. T. Jones, Modern Subsonic Aerodynamics, pg. 36).
[Figure 3]
It is clear from this diagram that below Re's of about 60,000 a curved plate (e.g. a sail) gives the most lift, but by an Re of 120,000 a thick wing is superior.
The obvious question to ask is: What is the Reynolds number of the sail of a boat? Obviously, the bigger the sail, the bigger the Re. Let's look into a very small 3-meter (10 foot) sailing dingy, with a 6-meter tall (20 foot) sail. What is worked out here for such a small craft applies with more force (literally) to even larger sailboats.
A PRACTICAL EXAMPLE
My friend, David Wing (really!) sails his beautifully home-made dinghies on the calm waters of San Diego bay. The size referred to in figuring out the Reynolds number is the length of the sail parallel to the wind direction. This length, in aerodynamic parlance, is called the chord of the sail (or airfoil). The chord at the pointy tip of a traditional triangular sail is nearly nothing, but the average chord — a bit lower than halfway down — is about 1.5 meters. The velocity is that of the wind relative to the foil when the boat is standing still. On a lovely San Diego day the wind is typically 3 to 6 meters/second. This gives us a typical Re of about
70,000 * 4 * 1.5
which works out to 420,000. This is above the Reynolds number where thin, curved airfoils work well; when the boat is under way, the wind relative to the sail is still faster, making Re larger still. The traditional cloth sail is superior — for this size boat — only when the wind is well below 1 m/sec. That's below 3 feet per second, or (to put it into other units) 2 miles per hour: a very light breeze indeed.
There is a figure of merit for airfoils, called the coefficient of lift, which we will notate with the abbreviation Cl. The amount of force you can get from a sail is proportional to Cl (all other things being equal). The very best a cloth sail can do is Cl = 0.8 or so (I am referring to a mainsail without a jib or other foresail). It will turn out that we can design a wing that will give twice as much push as this for the same (or less) weight and size.
So now let's imagine a wing standing up in the middle of the boat. Its chord at the bottom is 2 meters, its chord at the top 1 meter, and its height is 6 meters. Thus its average chord (1.5 meters) and height is the same as for the traditional sail, and therefore its area is the same as well: 9.0 square meters.
[Figure 4]
We have the chord and the height for the sail, but what should the cross-section look like? For this we look up books that allow us to choose good airfoils for the Reynolds number we need. One problem is that most airfoils are designed to lift an airplane, and they tend to work much better in lifting up than in lifting down (which is a good thing for the passengers). But our sailboat must be able to tack on both sides of the wind, and the airfoil must work equally well both ways. This means that unlike most airplane wings, the curve on both sides will be the same. Such airfoils are called "symmetrical" and that's what we need.
FINDING THE WING WE NEED
The trouble is that symmetrical airfoils don't have nearly as much lift, or force, to propel our boat, as asymmetrical airfoils do. The solution is to have a "flap" or adjustable portion of the airfoil that can bend like the wing flaps or ailerons on an airplane wing.
A bit of library research turns up a shape that seems like it was just made for us: it is designed to work with a flap and is symmetrical, and works best at Re of about 500,000 -- which is just what we need.
The shape looks like this
[Figure 5]
And was designed by the aerodynamicist Wortman to work at Re of 500,000 and to have the trailing 30% of it used as a flap (or aileron, considering how it tilts the boat). The accompanying table of coordinates can be used to scale it to any desired size. [Since this was written, research by my son and myself has resulted in symmetrical airfoils which may have even better performance.]
[Figure 6]
USING A WING ON THE WATER
Sailing with a wing is much like sailing with any other kind of sail. You will trim the wing angle for best oomph and then add flap while decreasing the angle until a stall's burble (as shown by small strips of mylar or yarn attached to various places on the wing) is just felt or seen and then going back a bit. Unlike a traditional sail, it would be possible to set up a wing so that almost no effort is required to gibe, but for a small boat the advantages of having the sail pivoted aft of the mast so that the mast can be guyed cannot be gainsaid.
Earlier we noted that with a conventional sail, a Cl of anything over 1 would be hard to achieve (with best performance coming at .8), but with a wing a Cl over 1.5 is possible. We get nearly twice the effectiveness per area with a wing as compared to a traditional sail. No wonder Sail America choose rigid wings.
WELL, JUST HOW MUCH OOMPH WILL IT HAVE?
Let's see just how much force we get. The force (in Newtons) generated by a wing or sail is
Cl * .5 * density of air * velocity squared * area.
With a mere breeze of 3 meters per second we get a force of
.8 * .5 * 1.225 * 3 * 3 * 9
or about 40 Newtons for the traditional sail. Divide Newtons by 2 to get (within 10 per cent or so) pounds force.
Now with the wing, we get
1.5 * .5 * 1.225 * 3 * 3 * 9
or 74 Newtons.
With less gentle zephyrs and with the flap bent we could easily obtain a force of 200 to 250 Newtons, which is a lot of cookies.
NO CORRECTIONS NEEDED FOR ALTITUDE
Due to a very strange coincidence, I have observed most boats are operated at or about sea level (Sailors on lake Titicaca can ignore this paragraph). This means that, unlike working with aircraft, we do not have to compensate for altitude in our calculations. Also, temperatures much below freezing are not often encountered (except, perhaps, by ice boats).
A foiled boat does not want to run dead before the wind (a foil turned sideways is no spinnaker, although a flapped section might catch some wind). Tacking downwind will be de rigeur.
Construction would not be difficult. It is a model airplane wing writ large, or a full size wing scaled down. Either the modern method of using foam core with fiberglass, or the traditional (for planes) ribs and spars covered with Monokote (a product made for model airplanes) would probably do. Such a wing can be lighter (!) than a cloth sail of the same area, and considerably lighter than a cloth sail of equivalent effectiveness. The available transparent Monokote would do fine for a see-through panel in the sail.
WHILE WE'RE AT IT, LET'S BUILD SOME OTHER FOILS
The keelboard of small boats should also be optimized. The density of water is huge compared to air: 1000 kg/cubic meter (salt water is a bit denser at 1025 kg/cubic meter) as opposed to the density of air, which is 1.225 kg/cubic meter. On the other hand the viscosity of water is much higher (water is thicker than air). It is found that the Re of an object in the water is roughly 860,000 * Velocity * Length.
If the boat's speed through the water is 1 meter/sec, and the keelboard has a chord of .5 meter, it would have a Re of 430,000. This is about the same as for our sail! So the traditional flat-plate keelboard is bad news indeed.
We could use the same cross-section as our sail, but in this case, we can observe nature and guess that the laminar NACA 63A016 must be close to right. Why the 63A016? Because it is almost identical to the cross section of fish that has a Re similar to your keelboard (The fish measurements are from Aeronautical and Miscellaneous Note-Book (ca. 1799-1826) of Sir George Cayley quoted in Theodore von Karman's classic Aerodynamics, 1954, pg. 8 where he compares Cayley's measurements to the airfoil.) Experiment also suggests that having the thickest point of the foil about halfway back delays cavitation at high coefficients of lift, when the keelboard is working hardest. You don't get cavitation in air, but we do have to worry about it here.
[Figure 6, fish and NACA 63A016]
David Wing, referred to above, made this one change to his Melody and reports that for the first time, the boat planed: it was the diminished drag of the keelboard that made the difference. He tells me that the Melody class of boats don't plane as a rule.
A flapped keelboard with the aft portion movable as on the wing, (a little lever protruding from the keelboard housing would be a new control!), would probably much improve the performance. This is not a totally new idea; it has been tried and it works. Since the side force generated by the proper foil can be much larger than that of a flat plate, the flapped keelboard could be reduced almost to half the size and weight of the ones now being used, or the same size keelboard could be much more effective with less drag. In any case, the boat would go faster, and heat up the bay's waters less if the keelboard were properly shaped.
[Figure 8, Keelboard]
A flapped, properly shaped rudder would also give the needed turning forces with less drag than the usual shape.
[Figure 9, Rudder]
A TALL TALE
Tall, thin sails are said to have a high aspect ratio. Short, squat ones have a low aspect ratio. Aspect ratio is obtained by dividing the height by the average chord. In our dingy, the aspect ratio is about 4. It turns out that given equal areas, the higher the aspect ratio of a sail, the more efficient it is (this is true for traditional sails and wings alike, but false for small, model boats). This effect is most marked at high coefficients of lift, where you will be sailing. The curve of improvement of efficiency is rather steep until it reaches a knee at an aspect ratio of about 5, after which small increases in aspect ratio yield ever diminishing returns. If there is no airflow under the sail, then the wing is "reflected" by the water in an aerodynamic sense, and the geometric aspect ratio is doubled to become the effective aspect ratio. The water becomes a large end-plate. The trouble in realizing this goal seems to be that you want to avoid having to leave the boat every time you come about. In any case, the smaller the gap between sail and gunwale, the more effective the sail.
Until you can no longer get it strapped on top of your Volvo, the taller the sail the more efficient it will be before diminishing returns (and likelihood of capsizing) take controlling effect. I say "diminishing returns" since increasing the aspect ratio by merely lowering the chord reduces the Reynolds number, thus making it less efficient. This effect is only pronounced if the Reynolds number falls below 250,000 or so, larger boats can ignore it.
WHY YOU SHOULD GET HIGH
The taller the sail, the more powerful the winds it reaches. This phenomenon is called "wind shear" and reflects the fact that the closer to the surface you are, the slower the wind is going. There is a little-known formula for this effect. If you measure the windspeed as w1 at a height of z1 above the surface, then the windspeed w2 at a height of z2 (on an essentially flat surface such as the bay or the Asian steppes) can be found from the empirical relationship w2 = w1 * (z2/z1)^(1/7). For example, if the wind is 3 m/sec at 1.5 meter eye-height then at the top of a four meter sail, it is whistling by at (2.67) ^ (.14) = 1.15 * 3 or about 3.5 meters/second. Not much to write home about. You get double the 5 foot level airspeed at 120 feet or so, which does interest larger craft. Small is beautiful, but with aircraft and boats, it is large that is most efficient. Too bad, I like small. But even small, the wing's the thing.
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